If x=4 and y=1, find the value of `((x)/(2)-(y)/(3))((x^(2))/(4)+(xy)/(6)+(y^(2))/(9))`.

Answer

To solve the expression \(\left(\frac{x}{2} - \frac{y}{3}\right)\left(\frac{x^2}{4} + \frac{xy}{6} + \frac{y^2}{9}\right)\) given \(x = 4\) and \(y = 1\), we can follow these steps:

### Step 1: Substitute the values of \(x\) and \(y\)
Substituting \(x = 4\) and \(y = 1\) into the expression:
\[ \left(\frac{4}{2} - \frac{1}{3}\right)\left(\frac{4^2}{4} + \frac{4 \cdot 1}{6} + \frac{1^2}{9}\right) \]

### Step 2: Simplify the first part \(\frac{4}{2} - \frac{1}{3}\)
Calculating \(\frac{4}{2}\):
\[ \frac{4}{2} = 2 \]
Now, calculating \(2 - \frac{1}{3}\):
\[ 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3} \]

### Step 3: Simplify the second part \(\frac{4^2}{4} + \frac{4 \cdot 1}{6} + \frac{1^2}{9}\)
Calculating each term:
\[ \frac{4^2}{4} = \frac{16}{4} = 4 \]
\[ \frac{4 \cdot 1}{6} = \frac{4}{6} = \frac{2}{3} \]
\[ \frac{1^2}{9} = \frac{1}{9} \]
Now, combine these results:
\[ 4 + \frac{2}{3} + \frac{1}{9} \]
To add these, we need a common denominator. The least common multiple of 1, 3, and 9 is 9:
\[ 4 = \frac{36}{9}, \quad \frac{2}{3} = \frac{6}{9}, \quad \frac{1}{9} = \frac{1}{9} \]
Now, adding them together:
\[ \frac{36}{9} + \frac{6}{9} + \frac{1}{9} = \frac{36 + 6 + 1}{9} = \frac{43}{9} \]

### Step 4: Multiply the two simplified parts
Now we have:
\[ \left(\frac{5}{3}\right)\left(\frac{43}{9}\right) \]
Multiplying these fractions:
\[ \frac{5 \cdot 43}{3 \cdot 9} = \frac{215}{27} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{215}{27} \]

---